Partially ordered group

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title: "Partially ordered group" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["ordered-algebraic-structures", "ordered-groups", "order-theory"] description: "Group with a compatible partial order" topic_path: "general/ordered-algebraic-structures" source: "https://en.wikipedia.org/wiki/Partially_ordered_group" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Group with a compatible partial order ::

In abstract algebra, a partially ordered group is a group (G, +) equipped with a partial order "≤" that is translation-invariant; in other words, "≤" has the property that, for all a, b, and g in G, if a ≤ b then a + g ≤ b + g and g +* a* ≤ g +* b*.

An element x of G is called positive if 0 ≤ x. The set of elements 0 ≤ x is often denoted with G+, and is called the '*positive cone of *G'''''.

By translation invariance, we have a ≤ b if and only if 0 ≤ -a + b. So we can reduce the partial order to a monadic property: if and only if

For the general group G, the existence of a positive cone specifies an order on G. A group G is a partially orderable group if and only if there exists a subset H (which is G+) of G such that:

• 0 ∈ H
• if a ∈ H and b ∈ H then a + b ∈ H
• if a ∈ H then -x + a + x ∈ H for each x of G
• if a ∈ H and -a ∈ H then a = 0

A partially ordered group G with positive cone G+ is said to be unperforated if n · g ∈ G+ for some positive integer n implies g ∈ G+. Being unperforated means there is no "gap" in the positive cone G+.

If the order on the group is a linear order, then it is said to be a linearly ordered group. If the order on the group is a lattice order, i.e. any two elements have a least upper bound, then it is a lattice-ordered group (shortly l-group, though usually typeset with a script l: ℓ-group).

A Riesz group is an unperforated partially ordered group with a property slightly weaker than being a lattice-ordered group. Namely, a Riesz group satisfies the Riesz interpolation property: if x1, x2, y1, y2 are elements of G and xi ≤ yj, then there exists z ∈ G such that xi ≤ z ≤ yj.

If G and H are two partially ordered groups, a map from G to H is a morphism of partially ordered groups if it is both a group homomorphism and a monotonic function. The partially ordered groups, together with this notion of morphism, form a category.

Partially ordered groups are used in the definition of valuations of fields.

Examples

• The integers with their usual order
• An ordered vector space is a partially ordered group
• A Riesz space is a lattice-ordered group
• A typical example of a partially ordered group is Zn, where the group operation is componentwise addition, and we write (a1,...,*an) ≤ (b1,...,*bn*) if and only if *ai* ≤ *b*i* (in the usual order of integers) for all i = 1,..., n.
• More generally, if G is a partially ordered group and X is some set, then the set of all functions from X to G is again a partially ordered group: all operations are performed componentwise. Furthermore, every subgroup of G is a partially ordered group: it inherits the order from G.
• If A is an approximately finite-dimensional C*-algebra, or more generally, if A is a stably finite unital C*-algebra, then K0(A) is a partially ordered abelian group. (Elliott, 1976)

Properties

Archimedean

The Archimedean property of the real numbers can be generalized to partially ordered groups.

:Property: A partially ordered group G is called Archimedean when for any a, b \in G, if e \le a \le b and a^n \le b for all n \ge 1 then a=e. Equivalently, when a \neq e, then for any b \in G, there is some n\in \mathbb{Z} such that b .

Integrally closed

A partially ordered group G is called integrally closed if for all elements a and b of G, if a**n ≤ b for all natural n then a ≤ 1.

This property is somewhat stronger than the fact that a partially ordered group is Archimedean, though for a lattice-ordered group to be integrally closed and to be Archimedean is equivalent. There is a theorem that every integrally closed directed group is already abelian. This has to do with the fact that a directed group is embeddable into a complete lattice-ordered group if and only if it is integrally closed.

Note

References

• M. Anderson and T. Feil, Lattice Ordered Groups: an Introduction, D. Reidel, 1988.
• M. R. Darnel, The Theory of Lattice-Ordered Groups, Lecture Notes in Pure and Applied Mathematics 187, Marcel Dekker, 1995.
• L. Fuchs, Partially Ordered Algebraic Systems, Pergamon Press, 1963.
• V. M. Kopytov and A. I. Kokorin (trans. by D. Louvish), Fully Ordered Groups, Halsted Press (John Wiley & Sons), 1974.
• V. M. Kopytov and N. Ya. Medvedev, Right-ordered groups, Siberian School of Algebra and Logic, Consultants Bureau, 1996.
• R. B. Mura and A. Rhemtulla, Orderable groups, Lecture Notes in Pure and Applied Mathematics 27, Marcel Dekker, 1977.
• , chap. 9.

References

1. {{harvtxt. Birkhoff. 1942

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